ωn = √(k/m) • ζ = c/2√(km) • Free • Damped • Forced — Simulate • Explore • Practice • Quiz
Mechanical vibrations are oscillatory motions of bodies and structures that occur in virtually every engineering system. From the suspension of a car to the strings of a guitar, from earthquake-resistant buildings to precision machining, understanding vibrations is critical for mechanical, civil, and aerospace engineers. A spring-mass-damper system is the fundamental model used to study vibrations — it consists of a mass (m) connected to a spring (stiffness k) and a viscous damper (damping coefficient c). This simple yet powerful model captures the essential physics of oscillatory motion, including natural frequency, damping ratio, resonance, and frequency response.
Free undamped vibration occurs when a system oscillates at its natural frequency ωn = √(k/m) without energy dissipation — an idealised case producing perpetual sinusoidal motion x(t) = A·cos(ωn·t). In reality, all systems have some damping. Free damped vibration introduces the damping ratio ζ = c/(2√(km)), producing three distinct behaviours: underdamped (ζ < 1) with oscillatory decay, critically damped (ζ = 1) with the fastest non-oscillatory return to equilibrium, and overdamped (ζ > 1) with slow exponential return. Forced vibration occurs when an external periodic force drives the system. The steady-state response depends on the frequency ratio r = ω/ωn. Forced damped vibration combines both phenomena, with the amplitude magnification factor X = (F0/k)/√((1−r²)² + (2ζr)²) and phase angle φ = arctan(2ζr/(1−r²)).
Resonance occurs when the forcing frequency equals or approaches the natural frequency (ω ≈ ωn, r ≈ 1). At resonance, the amplitude of oscillation can grow dramatically — theoretically to infinity in an undamped system. This is why the Tacoma Narrows Bridge collapsed in 1940 and why soldiers break step when crossing bridges. Damping limits the peak amplitude at resonance: the lower the damping ratio, the sharper and higher the resonance peak. Engineers must design systems to either avoid resonance or provide sufficient damping to limit dangerous vibration amplitudes. Vibration isolation, achieved by choosing system parameters so that r > √2, ensures transmitted force is less than the applied force.
In Simulate mode, select a vibration type (Free Undamped, Free Damped, Forced, or Forced Damped) and adjust mass, spring constant, damping coefficient, forcing frequency, forcing amplitude, and initial displacement. The left side of the canvas shows an animated spring-mass-damper system with a realistic zigzag spring, dashpot, and oscillating mass. The right side displays a live waveform chart (oscilloscope-style) showing displacement vs time with envelope curves for damped oscillations. Readout cards display natural frequency, damping ratio, period, amplitude, phase angle, and frequency ratio in real time. Use presets like Car Suspension, Bridge Resonance, Tuning Fork, and Earthquake Damper to explore realistic configurations. Switch to Explore to study 12 vibration concepts across Free Vibrations, Forced Vibrations, and Applications. Practice generates random calculation problems, and Quiz tests your knowledge with 5 randomised questions.
The natural frequency ωn = √(k/m) determines how fast a system oscillates when released from displacement. The damping ratio ζ = c/(2mωn) classifies the system response. The damped natural frequency ωd = ωn√(1−ζ²) is the actual oscillation frequency of an underdamped system. The logarithmic decrement δ = ln(xn/xn+1) = 2πζ/√(1−ζ²) quantifies the rate of amplitude decay. For forced systems, the transmissibility ratio and magnification factor are essential for vibration isolation design. All these calculations are performed in real time by this simulator.
This simulator serves mechanical engineering students studying vibrations and dynamics, automotive engineers designing suspension systems, civil engineers analysing structural dynamics, aerospace engineers studying flutter and aeroelastic phenomena, physics students learning about oscillatory motion, and instructors teaching mechanical vibrations or dynamics of machinery. It provides hands-on visual understanding without laboratory equipment or commercial software.